In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
is a Puiseux series in the indeterminate . Puiseux series were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.Puiseux (1850, 1851)
The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator , a Puiseux series becomes a Laurent series in an th root of the indeterminate. For example, the example above is a Laurent series in Because a complex number has th roots, a convergent Puiseux series typically defines functions in a neighborhood of .
Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation with complex coefficients, its solutions in , viewed as functions of , may be expanded as Puiseux series in that are convergent in some neighbourhood of . In other words, every branch of an algebraic curve may be locally described by a Puiseux series in (or in when considering branches above a neighborhood of ).
Using modern terminology, Puiseux's theorem asserts that the set of Puiseux series over an algebraically closed field of characteristic 0 is itself an algebraically closed field, called the field of Puiseux series. It is the algebraic closure of the field of formal Laurent series, which itself is the field of fractions of the ring of formal power series.
Definition If is a field (such as the complex numbers), a Puiseux series with coefficients in is an expression of the form
where is a positive integer and is an integer. In other words, Puiseux series differ from Laurent series in that they allow for fractional exponents of the indeterminate, as long as these fractional exponents have bounded denominator (here n). Just as with Laurent series, Puiseux series allow for negative exponents of the indeterminate as long as these negative exponents are bounded below (here by ). Addition and multiplication are as expected: for example,
and
One might define them by first "upgrading" the denominator of the exponents to some common denominator and then performing the operation in the corresponding field of formal Laurent series of .
The Puiseux series with coefficients in form a field, which is the union
of fields of formal Laurent series in (considered as an indeterminate).
This yields an alternative definition of the field of Puiseux series in terms of a direct limit. For every positive integer , let be an indeterminate (meant to represent ), and be the field of formal Laurent series in If divides , the mapping induces a field homomorphism and these homomorphisms form a direct system that has the field of Puiseux series as a direct limit. The fact that every field homomorphism is injective shows that this direct limit can be identified with the above union, and that the two definitions are equivalent (up to a canonical isomorphism).
Valuation A nonzero Puiseux series can be uniquely written as
with The valuation
of is the smallest exponent for the natural order of the rational numbers, and the corresponding coefficient is called the initial coefficient or valuation coefficient of . The valuation of the zero series is
The function is a valuation and makes the Puiseux series a valued field, with the additive group of the rational numbers as its valuation group.
As for every valued fields, the valuation defines a ultrametric distance by the formula For this distance, the field of Puiseux series is a metric space. The notation
expresses that a Puiseux is the limit of its partial sums. However, the field of Puiseux series is not complete; see below .
Convergent Puiseux series Puiseux series provided by Newton–Puiseux theorem are convergent in the sense that there is a neighborhood of zero in which they are convergent (0 excluded if the valuation is negative). More precisely, let
be a Puiseux series with complex coefficients. There is a real number , called the radius of convergence such that the series converges if is substituted for a nonzero complex number of absolute value less than , and is the largest number with this property. A Puiseux series is convergent if it has a nonzero radius of convergence.
Because a nonzero complex number has th roots, some care must be taken for the substitution: a specific th root of , say , must be chosen. Then the substitution consists of replacing by for every .
The existence of the radius of convergence results from the similar existence for a power series, applied to considered as a power series in
It is a part of Newton–Puiseux theorem that the provided Puiseux series have a positive radius of convergence, and thus define a (multivalued) analytic function in some neighborhood of zero (zero itself possibly excluded).
Valuation and order on coefficients If the base field is ordered, then the field of Puiseux series over is also naturally (“lexicographically”) ordered as follows: a non-zero Puiseux series with 0 is declared positive whenever its valuation coefficient is so. Essentially, this means that any positive rational power of the indeterminate is made positive, but smaller than any positive element in the base field .
If the base field is endowed with a valuation , then we can construct a different valuation on the field of Puiseux series over by letting the valuation be where is the previously defined valuation ( is the first non-zero coefficient) and is infinitely large (in other words, the value group of is ordered lexicographically, where is the value group of ). Essentially, this means that the previously defined valuation is corrected by an infinitesimal amount to take into account the valuation given on the base field.
Newton–Puiseux theorem As early as 1671,Newton (1736) Isaac Newton implicitly used Puiseux series and proved the following theorem for approximating with series the roots of algebraic equations whose coefficients are functions that are themselves approximated with series or polynomials. For this purpose, he introduced the Newton polygon, which remains a fundamental tool in this context. Newton worked with truncated series, and it is only in 1850 that Victor Puiseux introduced the concept of (non-truncated) Puiseux series and proved the theorem that is now known as Puiseux's theorem or Newton–Puiseux theorem.cf. Kedlaya (2001), introduction The theorem asserts that, given an algebraic equation whose coefficients are polynomials or, more generally, Puiseux series over a field of characteristic zero, every solution of the equation can be expressed as a Puiseux series. Moreover, the proof provides an algorithm for computing these Puiseux series, and, when working over the complex numbers, the resulting series are convergent.
In modern terminology, the theorem can be restated as: the field of Puiseux series over an algebraically closed field of characteristic zero, and the field of convergent Puiseux series over the complex numbers, are both algebraically closed.
Newton polygon
Let
be a polynomial whose nonzero coefficients are polynomials, power series, or even Puiseux series in . In this section, the valuation of is the lowest exponent of in (Most of what follows applies more generally to coefficients in any valued ring.)
For computing the Puiseux series that are roots of (that is solutions of the functional equation ), the first thing to do is to compute the valuation of the roots. This is the role of the Newton polygon.
Let us consider, in a Cartesian plane, the points of coordinates The Newton polygon of is the lower convex hull of these points. That is, the edges of the Newton polygon are the line segments joining two of these points, such that all these points are not below the line supporting the segment (below is, as usually, relative to the value of the second coordinate).
Given a Puiseux series of valuation , the valuation of is at least the minimum of the numbers and is equal to this minimum if this minimum is reached for only one . So, for being a root of , the minimum must be reached at least twice. That is, there must be two values and of such that and for every .
That is, and must belong to an edge of the Newton polygon, and
must be the opposite of the slope of this edge. This is a rational number as soon as all valuations are rational numbers, and this is the reason for introducing rational exponents in Puiseux series.
In summary, the valuation of a root of must be the opposite of a slope of an edge of the Newton polygon.
The initial coefficient of a Puiseux series solution of can easily be deduced. Let be the initial coefficient of that is, the coefficient of in Let be a slope of the Newton polygon, and be the initial term of a corresponding Puiseux series solution of If no cancellation would occur, then the initial coefficient of would be where is the set of the indices such that belongs to the edge of slope of the Newton polygon. So, for having a root, the initial coefficient must be a nonzero root of the polynomial
(this notation will be used in the next section).
In summary, the Newton polygon allows an easy computation of all possible initial terms of Puiseux series that are solutions of
The proof of Newton–Puiseux theorem will consist of starting from these initial terms for computing recursively the next terms of the Puiseux series solutions.
Constructive proof Let suppose that the first term of a Puiseux series solution of has been be computed by the method of the preceding section. It remains to compute For this, we set and write the Taylor expansion of at
This is a polynomial in whose coefficients are Puiseux series in . One may apply to it the method of the Newton polygon, and iterate for getting the terms of the Puiseux series, one after the other. But some care is required for insuring that and showing that one get a Puiseux series, that is, that the denominators of the exponents of remain bounded.
The derivation with respect to does not change the valuation in of the coefficients; that is,
and the equality occurs if and only if where is the polynomial of the preceding section. If is the multiplicity of as a root of it results that the inequality is an equality for The terms such that can be forgotten as far as one is concerned by valuations, as and imply
This means that, for iterating the method of Newton polygon, one can and one must consider only the part of the Newton polygon whose first coordinates belongs to the interval Two cases have to be considered separately and will be the subject of next subsections, the so-called ramified case, where , and the regular case where .
Ramified case The way of applying recursively the method of the Newton polygon has been described precedingly. As each application of the method may increase, in the ramified case, the denominators of exponents (valuations), it remains to prove that one reaches the regular case after a finite number of iterations (otherwise the denominators of the exponents of the resulting series would not be bounded, and this series would not be a Puiseux series. By the way, it will also be proved that one gets exactly as many Puiseux series solutions as expected, that is the degree of in .
Without loss of generality, one can suppose that that is, Indeed, each factor of provides a solution that is the zero Puiseux series, and such factors can be factored out.